Modules and rings

This Book On Modern And Non-commutative Ring Theory Is Ideal For Beginning Graduate Students. It Starts At The Foundations Of The Subject And Progresses Rapidly Through The Basic Concepts To Help The Reader Reach Current Research Frontiers. Students Will Have The Chance To Develop Proofs, Solve Problems, And To Find Interesting Questions. The First Half Of The Book Is Concerned With Free, Projective And Injective Modules, Tensor Algebras, Simple Modules And Primitive Rings, The Jacobson Radical, And Subdirect Products. Later In The Book, More Advanced Topics, Such As Hereditary Rings, Categories And Functors, Flat Modules And Purity Are Introduced. These Later Chapters Will Also Prove A Useful Reference For Researchers In Non-commutative Ring Theory. Enough Background Material (including Detailed Proofs) Is Supplied To Give The Student A Firm Grounding In The Subject. Modules -- Free Modules -- Injective Modules -- Tensor Products -- Certain Important Algebras -- Simple Modules And Primitive Rings -- The Jacobson Radical -- Subdirect Product Decompositions -- Primes And Semiprimes -- Projective Modules And More On Wedderburn Theorems -- Direct Sum Decompositions -- Simple Algebras -- Hereditary Rings, Free And Projective Modules -- Module Constructions -- Categories And Functors -- Module Categories -- Flat Modules -- Purity. John Dauns. Includes Bibliographical References (p. 432-435) And Indexes. Cover......Page 1 Title page......Page 3 Copyright page......Page 4 Contents......Page 5 PREFACE......Page 11 NOTE TO THE READER......Page 17 1-1 Definitions......Page 19 1-2 Direct Products and Sums......Page 23 1-3 Adjunction of 1 to $R$......Page 26 1-4 Sequences of Modules......Page 28 1-5 Exercises......Page 30 Introduction......Page 37 2-1 Definition of Free Modules......Page 38 2-2 Bases of Free Modules......Page 42 2-3 Exercises......Page 47 3-1 Properties of Injectives......Page 48 3-2 Divisibility......Page 51 3-3 Embeddings in Injectives......Page 54 3-4 Injective Hulls......Page 57 3-5 Noetherian Rings......Page 63 3-6 Examples......Page 65 3-7 Exercises......Page 69 4-1 Tensor Products of Modules......Page 71 4-2 Definitions for Algebras......Page 78 4-3 Tensor Products of Algebras......Page 81 4-4 Exercises......Page 87 Introduction......Page 89 5-1 Free and Tensor Algebras......Page 90 5-2 Exterior Algebras......Page 92 5-3 Exercises......Page 101 Introduction......Page 104 6-1 Preliminaries......Page 105 6-2 Cyclic Modules......Page 108 6-3 Simple Modules......Page 109 6-4 Examples......Page 112 6-5 Density......Page 113 6-6 More on Density and Simples......Page 117 6-7 Examples......Page 123 6-8 Exercises......Page 127 Introduction......Page 129 7-1 Characterizations......Page 130 7-2 Radicals of Related Rings......Page 143 7-3 Local Rings......Page 150 7-4 Examples......Page 153 7-5 Exercises......Page 156 Introduction......Page 158 8-1 Subdirect Products......Page 159 8-2 Dense Subdirect Products......Page 163 8-3 Exercises......Page 165 Introduction......Page 166 9-1 Prime Ideals......Page 167 9-2 Semiprime Ideals and the Prime Radical......Page 169 9-3 Nil Radicals......Page 174 9-4 Primes and Semiprimes in Derived Rings......Page 176 9-5 Exercises......Page 180 Introduction......Page 181 10-1 Projective Modules......Page 183 10-2 Projective Dimension......Page 190 10-3 Minimal Right Ideals......Page 195 10-4 Main Theorems......Page 198 10-5 Direct Proofs......Page 202 10-6 Uniqueness......Page 208 10-7 Rings with D.C.C. and Idempotents......Page 209 10-8 Exercises......Page 214 Introduction......Page 222 11-1 Completely Reducible Modules......Page 224 11-2 Radical of a Module......Page 227 11-3 Artinian and Noetherian Modules......Page 231 11-4 Direct Sums of Indecomposables......Page 239 11-5 Singular Submodule......Page 249 11-6 Exercises......Page 252 Introduction......Page 257 12-1 Algebra Modules......Page 258 12-2 Multiplication Algebra......Page 259 12-3 Tensor Products of Simple Rings......Page 264 12-4 Centralizers......Page 268 12-5 Double Centralizers......Page 277 12-6 Exercises......Page 284 13-1 Hereditary Rings......Page 287 13-2 Injectivity and Projectivity......Page 290 13-3 Finitely Generated Modules......Page 293 13-4 Examples......Page 297 13-5 Exercises......Page 299 Introduction......Page 301 14-1 Pullbacks......Page 302 14-2 Pushouts......Page 307 14-3 Pushout Application......Page 311 14-4 Exercises......Page 313 15-1 Basics of Categories......Page 316 15-2 Objects......Page 330 15-3 Pre-additive Categories......Page 333 15-4 Adjoint Functors......Page 343 15-5 Exercises......Page 351 16-1 Generators and Cogenerators......Page 353 16-2 Horn Functor......Page 356 16-3 Tensor Product Functor......Page 357 16-4 Adjoint Associativity......Page 360 16-5 Elements of Tensor Products......Page 363 16-6 Direct and Inverse Limits......Page 365 16-7 Exercises......Page 370 16-8 Exercises on direct and inverse limits......Page 373 17-1 Character Modules......Page 377 17-2 Flat Module Basics......Page 380 17-3 Exercises......Page 383 Introduction......Page 385 18-1 Systems of Equations in Modules......Page 386 18-2 Pure Projectives and Pure Exact Sequences......Page 388 18-3 Direct Limits......Page 397 18-4 Pure Injectives......Page 401 18-5 Pure Injective Hull......Page 409 18-6 Exercises......Page 414 A-1 Sets, Symbols, and Functions......Page 416 A-2 Background Review......Page 423 A-3 Exercises......Page 426 B-2 Exterior Algebras......Page 430 B-3 A Unified Approach......Page 433 LIST OF SYMBOLS AND NOTATION......Page 445 BIBLIOGRAPHY......Page 450 SUBJECT INDEX......Page 454 AUTHOR INDEX......Page 460 This book on modern module and non-commutative ring theory is ideal for beginning graduate students. It starts at the foundations of the subject and progresses rapidly through the basic concepts to help the reader reach current research frontiers. Students will have the chance to develop proofs, solve problems, and to find interesting questions. The first half of the book is concerned with free, projective, and injective modules, tensor algebras, simple modules and primitive rings, the Jacobson radical, and subdirect products. Later in the book, more advanced topics, such as hereditary rings, categories and functors, flat modules, and purity are introduced. These later chapters will also prove a useful reference for researchers in non-commutative ring theory. Enough background material (including detailed proofs) is supplied to give the student a firm grounding in the subject. An ideal book on module and non-commutative ring theory for graduate students. It starts at the foundations of the subject and progresses rapidly through the basic concepts to help the reader reach current research frontiers.